In this paper, we introduce a class of Locally Adaptive Sampling schemes. In this sampling family, time intervals between samples can be computed by using a function of previously taken samples, called a sampling function. Hence, though it is a non-uniform sampling scheme, we do not need to keep sampling times. The aim of LAS is to have the average sampling rate and the reconstruction error satisfy some requirements. We propose four different schemes of LAS. The first two are designed for deterministic signals. First, we derive a Taylor Series Expansion (TSE) sampling function, which only assumes the third derivative of the signal is bounded, but requires no other specific knowledge of the signal. Then, a Discrete Time-Valued (DTV) sampling function is proposed, where the sampling time intervals are chosen from a lattice. Next, we consider stochastic signals. We propose two sampling methods based on linear prediction filters: a Generalized Linear Prediction (GLP) sampling function, and a Linear Prediction sampling function with Side Information (LPSI). In GLP method, we only assume the signal is locally stationary. However, LPSI is specifically designed for a known signal model.